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2009 | 7 | 3 | 365-381
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Periodic harmonic functions on lattices and points count in positive characteristic

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EN
This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.
Twórcy
Bibliografia
  • [1] Amin A.T., Slater P.J., Zhang G.-H., Parity dimension for graphs-a linear algebraic approach, Linear and Multilinear Algebra, 2002, 50, 327–342 http://dx.doi.org/10.1080/0308108021000049293
  • [2] Barua R., Ramakrishnan S., σ-game, σ+-game and two-dimensional additive cellular automata, Theoret. Comput. Sci., 1996, 154, 349–366 http://dx.doi.org/10.1016/0304-3975(95)00091-7
  • [3] Bhargava M., Zieve M.E., Factoring Dickson polynomials over finite fields, Finite Fields Appl., 1999, 5, 103–111 http://dx.doi.org/10.1006/ffta.1998.0221
  • [4] Bicknell M., A primer for the Fibonacci numbers VII, Fibonacci Quart., 1970, 8, 407–420
  • [5] Goldwasser J., Klostermeyer W., Ware H., Fibonacci Polynomials and Parity Domination in Grid Graphs, Graphs Combin., 2002, 18, 271–283 http://dx.doi.org/10.1007/s003730200020
  • [6] Goldwasser J., Wang X., Wu Y., Does the lit-only restriction make any difference for the σ-game and σ+-game?, European J. Combin., 2009, 30, 774–787 http://dx.doi.org/10.1016/j.ejc.2008.09.020
  • [7] Gravier S., Mhalla M., Tannier E., On a modular domination game, Theoret. Comput. Sci., 2003, 306, 291–303 http://dx.doi.org/10.1016/S0304-3975(03)00285-8
  • [8] Heath-Brown D.R., Artin’s conjecture for primitive roots, Quart. J. Math., 1986, 37, 27–38 http://dx.doi.org/10.1093/qmath/37.1.27
  • [9] Hoggatt V.E.Jr., Bicknell-Johnson M., Divisibility properties of polynomials in Pascal’s triangle, Fibonacci Quart., 1978, 16, 501–513
  • [10] Hoggatt V.E.Jr., Long C.T., Divisibility properties of generalized Fibonacci polynomials, Fibonacci Quart., 1974, 12, 113–120
  • [11] Humphreys J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1978
  • [12] Hunziker M., Machiavelo A., Park J., Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids, Theoret. Comput. Sci., 2004, 320, 465–483 http://dx.doi.org/10.1016/j.tcs.2004.03.031
  • [13] Jacob G., Reutenauer C., Sakarovitch J., On a divisibility property of Fibonacci polynomials, preprint available at http://en.scientificcommons.org/43936584
  • [14] Levy D., The irreducible factorization of Fibonacci polynomials over Q, Fibonacci Quart., 2001, 39, 309–319
  • [15] Lidl R., Mullen G.L., Turnwald G., Dickson polynomials, Longman Scientific and Technical, Harlow, John Wiley and Sons, Inc., New York, 1993
  • [16] Martin O., Odlyzko A.M., Wolfram S., Algebraic properties of cellular automata, Comm. Math. Phys., 1984, 93, 219–258 http://dx.doi.org/10.1007/BF01223745
  • [17] Moree P., Artin’s primitive root conjecture-a survey, preprint available at http://arxiv.org/abs/math/0412262
  • [18] Ram Murty M., Artin’s conjecture for primitive roots, Math. Intelligencer, 1988, 10, 59–67 http://dx.doi.org/10.1007/BF03023749
  • [19] Sarkar P., Barua R., Multidimensional σ-automata, π-polynomials and generalised S-matrices, Theoret. Comput. Sci., 1998, 197, 111–138 http://dx.doi.org/10.1016/S0304-3975(97)00160-6
  • [20] Sutner K., σ-automata and Chebyshev-polynomials, Theoret. Comput. Sci., 2000, 230, 49–73 http://dx.doi.org/10.1016/S0304-3975(97)00242-9
  • [21] Webb W.A., Parberry E.A., Divisibility properties of Fibonacci polynomials, Fibonacci Quart., 1969, 7, 457–463
  • [22] Zaidenberg M., Periodic binary harmonic functions on lattices, Adv. in Appl. Math., 2008, 40, 225–265 http://dx.doi.org/10.1016/j.aam.2007.01.004
  • [23] Zaidenberg M., Convolution equations on lattices: periodic solutions with values in a prime characteristic field, In: Kapranov M., Kolyada S., Manin Y.I., Moree P., Potyagailo L.A. (Eds.), Geometry and Dynamics of Groups and Spaces, In Memory of Alexander Reznikov, Progress in Mathematics 265, 719–740, Birkhäuser, 2008
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-009-0029-0
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